Optimal. Leaf size=72 \[ -\frac {4 \left (a x^3+b\right )^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^3+b}}{\sqrt [4]{2} x}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{a x^3+b}}\right ) \]
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Rubi [F] time = 3.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx &=\int \left (\frac {4 b}{x^4 \sqrt [4]{b+a x^3}}+\frac {a}{x \sqrt [4]{b+a x^3}}+\frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}\right ) \, dx\\ &=a \int \frac {1}{x \sqrt [4]{b+a x^3}} \, dx+(4 b) \int \frac {1}{x^4 \sqrt [4]{b+a x^3}} \, dx+\int \frac {4 b+a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^3\right )+\frac {1}{3} (4 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{b+a x}} \, dx,x,x^3\right )+\int \left (\frac {4 b}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}+\frac {a x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )}\right ) \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^3\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+\frac {2 a \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^3}\right )+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ &=-\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}+a \int \frac {x^3}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{b+a x^3} \left (b+a x^3-2 x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4 b+a x^3\right ) \left (-b-a x^3+x^4\right )}{x^4 \sqrt [4]{b+a x^3} \left (-b-a x^3+2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.91, size = 72, normalized size = 1.00 \begin {gather*} -\frac {4 \left (b+a x^3\right )^{3/4}}{3 x^3}-2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^3}}{\sqrt [4]{2} x}\right )+2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{b+a x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}+4 b \right ) \left (-a \,x^{3}+x^{4}-b \right )}{x^{4} \left (a \,x^{3}+b \right )^{\frac {1}{4}} \left (-a \,x^{3}+2 x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - x^{4} + b\right )} {\left (a x^{3} + 4 \, b\right )}}{{\left (a x^{3} - 2 \, x^{4} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,x^3+4\,b\right )\,\left (-x^4+a\,x^3+b\right )}{x^4\,{\left (a\,x^3+b\right )}^{1/4}\,\left (-2\,x^4+a\,x^3+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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